DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS

Transcription

1 DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper winess work, and resoluion in wrongful deah cases, requires esimaion of he amoun of loss. Tha loss resuls from loss of income ha he deceased would have earned had he/she remained alive. The crucial modeling problem is he uncerainy of he fuure income, and he change of i resuling from deah or injury. In his work, we apply he heory of sochasic processes o his problem. We consider he case of a single paymen, and a series of level paymens, corresponding o common mehods of paymens of awards in loss liigaion. We esablish he discouning procedure for he value of fuure salary, and apply sochasic simulaion o deermine loss in case of wrongful deah. We consider discouning wih a single ineres rae, and discouning when ineres raes are modeled wih a sochasic process. We also show ha he models can be exended o injury as well as oher liabiliy cases. INTRODUCTION Liigaion of wrongful deah and injury resuls in an imporan problem of deermining he loss amoun o be compensaed. The loss considered usually refers o income loss of affeced wrongful deah or injury persons minus his (or her) consumpion. Because he income loss of affeced persons is dynamic, ha is o say, he income changes over ime, we mus consider he dynamic change of ineres rae and salary income comprehensively. In his aricle, we deermine he loss compensaed by quaniy analysis and esablishing dynamic model of income loss of wrongful deah. I should be noiced ha i is an imporan work o analyze he changing paern of salary rae wih ime and forecas he income loss ha should be compensaed. There are several aricles ha have sudied his aspec. [2] proposed parameer esimaion of salary funcion, [5] proposed log wage equaion and [3] invesigaed income model of auo-correlaion Y() a + by( 1) + e where Y() is he income, a and b are consans and e is random error a ime. He poined ou ha here exised some limiaion for auo correlaion model, because many economic ime series are non-saionary. The resuls he analysis relaed change of he series daa of wage raes wih years suppored he assumpion of random walk model. Similar research was discussed by Horviz in [4]. He invesigaed he geomeric Brownian described by he following model Y() Y()e a 1 2 b2 +bz( ), where z() N(,). Anoher imporan issue is how o deermine he discoun value of income excep for forecasing he change paern of wage income wih he ime. In his aricle, we esablish dynamic compensaion models of wrongful deah based on he assuming ha discoun rae is a consan and a sochasic process respecively. We also assume ha he increase rae of wage is a sochasic process and here exiss correlaion beween he discoun rae and he increase rae of wage. The models esablished including he forms of a lump-sum paymen and level paymen. The calculaing mehods can be easily exended o he case of wrongful injury.

2 1. Models of a lump sum paymen Assuming ha he ineres rae is a consan Assume ha a person died in life hur a age. The compensaion paid o his legiimae successor should be equal o he presen value of his cumulaive income from he dae of his deah o he dae of his reiremen ha includes premium for reiremen annuiy minus his consumpion. Here we neglec he addiional paymen for pain and suffering o survivors. We also consider he probabiliy ha he dies in oher acciden or illness. Assume ha he increase rae of wage can be wrien as he following form [6]: dw W µ W d + σ W dz W (1) where µ W is he average wage rae, σ W is he sandard deviaion if wage rae and z W is a sandard Wiener process. Then we have W + µ Ws W s ds + σ Ws W s dz Ws (2) where W is he level of he wage of affeced person when wrongful deah even occurs. When µ W µ W and σ W σ W are consan he sochasic differenial equaion (1) has he unique soluion based on Io s Lemma W e µ W 1 2 σ 2 W +σ W z w (3) Assume ha he random variable T x expresses he survival ime if he affeced person aged x does no die in life hur. Le p x+ Pr ( T x < + 1) express he probabiliy ha a person aged x survives years. The values of p x+ can be found from he life ables. Here we also assume ha T x is a random variable ha is independen of he wage and he ineres rae. Assume ha he age of reiremen is rae is and he risk free ineres, hen he discoun facor calculaed by coninuous ineres rae can be expressed as. To simplify analysis, assume ha he consumpion for a person is h percen of his wage. Le P 1 be he presen value of compensaion of wrongful dearh wih a single paymen, hen P 1 E W (1 h) p x+ E where E denoes he expeced value. Le W (1 h)e(µ W 1 2 σ W 2 )+σ W z w p x+ e r W (1 h)eµ W p x+ e r, (4) e µ W p x+ e r k 1, (5) where k 1 is called as he facor of he compensaion of he wrongful deah wih a single paymen. By

4 Soluion of (13) is obained wih he help of Mone Carlo simulaion. (1 h)k 2 (13) 2. Models of compensaion for wrongful dearh wih level paymen in n years Because he way of a lump sum paymen needs obligaors o pay a lo of money one ime, someimes, i is difficul o redeem. One effecive mehod o solve his problem is o ake he form of periodical paymen. Assume ha he age he obligaor begins o pay he claim is y, p y+ expresses he probabiliy ha he obligaor aged survives years, and he value of periodical paymen paid a he beginning of each year is AP. Assuming he ineres rae is a consan The sum of presen value ha one dollar is paid a he beginning of each year wihin n years equals 1+ p y+ (14) Based on he principles of acuarial science, he value paid a he beginning of each year by he obligaor should be equal o he value paid a he form of a lump sum divided by he sum of presen value ha one dollar is paid a he beginning of each year wihin n years. Tha is AP 1 E W (1 h)e (µ W 1 2 σ W 1+ p y+ 2 )+σ W z w p x+ e r W (1 h)e µ W p x+ 1+ p y+ (15) Le e µ W p x+ 1+ p y+ k 3 (16) where k 3 is called as he facor of compensaion for he wrongful deah when he ineres rae is a consan. By combining equaion (15) and equaion (16), we obain AP 1 (1 h)k 3 (17) Assuming he ineres rae is a sochasic process The sum of presen value of one dollar paid a he beginning of each year wihin n years equals r u du} 1+ E p y+ e B( )r 1+ E p ( y+ A()e ) (18) Divided by equaion (18), he equaion (11) becomes

5 E A W (1 h)e (µ W 1 2 σ 2 W )+σ W z w B( )r p A()e x+ B( )r ( ) 1+ E p y+ A()e (19) Le E e (µ W 1 2 σ 2 W )+σ W z w B( )r p A()e x+ B( )r ( ) 1+ E p y+ A()e (2) where is called as he facor of compensaion for he wrongful deah when he ineres rae is a sochasic process. By combining equaion (19) wih equaion (2), we obained A (1 h). (21) 3. Calculaion of he compensaion for wrongful deah Assume β.1,σ r.15,σ W.1, µ r.35, µ W.1, ρ rw.2, 5, m 6. When he ineres rae is a consan, r.35. When he ineres rae is a sochasic process, he iniial level of he ineres rae r.3. The probabiliy ha he persons aged and survive years can be found from life ables. Table 1 and Figure 1 liss he values of he facors of he compensaion for he wrongful deah k 1,and calculaed wih he help of Mone Carlo simulaion and equaions of (5), (12), (16) and (2), where he deah age x Leing he iniial wage level when he affeced person dies W muliplied by (1 h) and by he facors of he compensaion for he wrongful deah k 1, and respecively, we can ge he values of he compensaion for he wrongful deah P 1,, AP 1 and A, where he meanings of k 1, and are illusraed in Table 2. Figure 2 is he descripion of he simulaion of sochasic processes when calculaing he facors of he compensaion for wrongful deah. Similarly, we can obain he values of he compensaion for he wrongful deah when he parameers ake oher values.

Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees

The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be

THE VALUATION AND HEDGING OF VARIABLE RATE SAVINGS ACCOUNTS BY FRANK DE JONG 1 AND JACCO WIELHOUWER ABSTRACT Variable rae savings accouns have wo main feaures. The ineres rae paid on he accoun is variable

LEASNG VERSUSBUYNG Conribued by James D. Blum and LeRoy D. Brooks Assisan Professors of Business Adminisraion Deparmen of Business Adminisraion Universiy of Delaware Newark, Delaware The auhors discuss

ECONOMIC AND BUSINESS REVIEW VOL. 13 No. 4 211 251 272 251 ADOPTION OF PROJECTED MORTALITY TABLE FOR THE SLOVENIAN MARKET USING THE POISSON LOG-BILINEAR MODEL TO TEST THE MINIMUM STANDARD FOR VALUING LIFE

Individual Healh Insurance April 30, 2008 Pages 167-170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve

Valuaion of Life Insurance Conracs wih Simulaed uaraneed Ineres Rae Xia uo and ao Wang Deparmen of Mahemaics Royal Insiue of echnology 100 44 Sockholm Acknowledgmens During he progress of he work on his

1 Time Consisency in Porfolio Managemen Traian A Pirvu Deparmen of Mahemaics and Saisics McMaser Universiy Torono, June 2010 The alk is based on join work wih Ivar Ekeland Time Consisency in Porfolio Managemen

Inroducion Chaper 14: Dynamic D-S dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuing-edge

VII. THE FIRM'S INVESTMENT DECISION UNDER CERTAINTY: CAPITAL BUDGETING AND RANKING OF NEW INVESTMENT PROJECTS The mos imporan decisions for a firm's managemen are is invesmen decisions. While i is surely

A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke

Chaper 6 Ineres Raes and Bond Valuaion Definiion and Descripion of Bonds Long-erm deb-loosely, bonds wih a mauriy of one year or more Shor-erm deb-less han a year o mauriy, also called unfunded deb Bond-sricly

UNIVERSITY OF CALGARY Modeling of Currency Trading Markes and Pricing Their Derivaives in a Markov Modulaed Environmen by Maksym Terychnyi A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL

Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions

QUADRATIC OPTIMIZATION OF LIFE AND PENSION INSURANCE PAYMENTS BY MOGENS STEFFENSEN ABSTRACT Quadraic opimizaion is he classical approach o opimal conrol of pension funds. Usually he paymen sream is approximaed

CLASSIFICATION OF REINSURANCE IN LIFE INSURANCE Kaarína Sakálová 1. Classificaions of reinsurance There are many differen ways in which reinsurance may be classified or disinguished. We will discuss briefly

Chaper 7. esponse of Firs-Order L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural

House Price Index (HPI) The price index of second hand houses in Colombia (HPI), regisers annually and quarerly he evoluion of prices of his ype of dwelling. The calculaion is based on he repeaed sales

Chaper Suden Lecure Noes - Chaper Goals QM: Business Saisics Chaper Analyzing and Forecasing -Series Daa Afer compleing his chaper, you should be able o: Idenify he componens presen in a ime series Develop

The Universiy of Liverpool School of Archiecure and Building Engineering WINDS PROJECT COURSE SYNTHESIS SECTION 3 UNIT 11 Marke Analysis and Models of Invesmen. Produc Developmen and Whole Life Cycle Cosing